Answer
$\frac{2}{3}(1+e^x)^\frac{3}{2}+C$
Work Step by Step
Let $u=1+e^x$. Then $du=e^x dx$.
$\int e^x\sqrt{1+e^x}dx$
$=\int \sqrt{1+e^x}e^xdx$
$=\int \sqrt{u}du$
$=\int u^\frac{1}{2}du$
$=\frac{u^\frac{3}{2}}{\frac{3}{2}}+C$
$=\frac{2}{3}u^\frac{3}{2}+C$
$=\frac{2}{3}(1+e^x)^\frac{3}{2}+C$
